Abstract

The problems encountered in the elaboration of measurements of direct and sky diffuse solar irradiance are the following: (1) to carry out the calibration for the direct irradiance, which consists in determining the direct irradiance at the upper limit of the atmosphere; (2) to carry out the calibration for the diffuse irradiance, which consists in determining the solid viewing angle of the sky radiometer; (3) to determine the input parameters, namely, ground albedo, real and imaginary parts of the aerosol refractive index, and aerosol radius range; and (4) to determine from the optical data the columnar aerosol optical depth and volume radius distribution. With experimental data and numerical simulations a procedure is shown that enables one to carry out the two calibrations needed for the sky radiometer, to determine a best estimate of the input parameters, and, finally, to obtain the average features of the atmospheric aerosols. An interesting finding is that inversion of only data of diffuse irradiance yields the same accuracy of result as data of both diffuse and direct irradiance; in this case, only calibration of the solid viewing angle of the sky radiometer is needed, thus shortening the elaboration procedure. Measurements were carried out in the Western Mediterranean Sea (Italy), in Tokyo (Japan), and in Ushuaia (Tierra del Fuego, Argentina); data were elaborated with a new software package, the Skyrad code, based on an efficient radiative transfer scheme.

© 1999 Optical Society of America

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References

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  1. D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
    [CrossRef]
  2. T. Nakajima, G. Tonna, R. Rao, P. Boi, Y. Kaufman, B. Holben, “Use of sky brightness measurements from ground for remote sensing of particulate polydispersions,” Appl. Opt. 35, 2672–2686 (1996).
    [CrossRef] [PubMed]
  3. Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
    [CrossRef]
  4. G. Dalu, R. Rao, A. Pompei, P. Boi, G. Tonna, B. Olivieri, “Aerosol optical properties retrieved from solar aureole measurements over southern Sardinia,” J. Geophys. Res. 100, 26,135–26,140 (1995).
    [CrossRef]
  5. G. Tonna, T. Nakajima, R. Rao, “Aerosol features retrieved from solar aureole data: a simulation study concerning a turbid atmosphere,” Appl. Opt. 34, 4486–4499 (1995).
    [CrossRef] [PubMed]
  6. T. Nakajima, M. Tanaka, T. Hayasaka, Y. Miyake, Y. Nakanishi, K. Sasamoto, “Airborne measurements of the optical stratification of aerosols in turbid atmospheres,” Appl. Opt. 25, 4374–4381 (1986).
    [CrossRef] [PubMed]
  7. F. Kasten, A. T. Young, “Revised optical air mass tables and approximation formula,” Appl. Opt. 28, 4735–4738 (1989).
    [CrossRef] [PubMed]
  8. C. Tomasi, E. Caroli, V. Vitale, “Study of the relationship between Ångström’s wavelength exponent and Junge particle size distribution exponent,” J. Clim. Appl. Meteorol. 22, 1707–1716 (1983).
    [CrossRef]
  9. M. Tanaka, T. Nakajima, M. Shiobara, “Calibration of a sunphotometer by simultaneous measurements of direct-solar and circumsolar radiations,” Appl. Opt. 25, 1170–1176 (1986).
    [CrossRef] [PubMed]
  10. M. D. King, B. M. Herman, “Determination of the ground albedo and the index of absorption of atmospheric particulates by remote sensing. I. Theory,” J. Atmos. Sci. 36, 163–173 (1979).
    [CrossRef]
  11. K. Krishna Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
    [CrossRef]
  12. J. A. Reagan, K. J. Thome, B. M. Herman, “A simple instrument and technique for measuring columnar water vapor via near-IR differential solar transmission measurements,” IEEE Trans. Geosci. Remote Sens. 30, 825–831 (1992).
    [CrossRef]
  13. U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
    [CrossRef]

1996 (2)

T. Nakajima, G. Tonna, R. Rao, P. Boi, Y. Kaufman, B. Holben, “Use of sky brightness measurements from ground for remote sensing of particulate polydispersions,” Appl. Opt. 35, 2672–2686 (1996).
[CrossRef] [PubMed]

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

1995 (2)

G. Dalu, R. Rao, A. Pompei, P. Boi, G. Tonna, B. Olivieri, “Aerosol optical properties retrieved from solar aureole measurements over southern Sardinia,” J. Geophys. Res. 100, 26,135–26,140 (1995).
[CrossRef]

G. Tonna, T. Nakajima, R. Rao, “Aerosol features retrieved from solar aureole data: a simulation study concerning a turbid atmosphere,” Appl. Opt. 34, 4486–4499 (1995).
[CrossRef] [PubMed]

1994 (1)

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

1992 (1)

J. A. Reagan, K. J. Thome, B. M. Herman, “A simple instrument and technique for measuring columnar water vapor via near-IR differential solar transmission measurements,” IEEE Trans. Geosci. Remote Sens. 30, 825–831 (1992).
[CrossRef]

1991 (1)

K. Krishna Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

1990 (1)

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

1989 (1)

1986 (2)

1983 (1)

C. Tomasi, E. Caroli, V. Vitale, “Study of the relationship between Ångström’s wavelength exponent and Junge particle size distribution exponent,” J. Clim. Appl. Meteorol. 22, 1707–1716 (1983).
[CrossRef]

1979 (1)

M. D. King, B. M. Herman, “Determination of the ground albedo and the index of absorption of atmospheric particulates by remote sensing. I. Theory,” J. Atmos. Sci. 36, 163–173 (1979).
[CrossRef]

Amato, U.

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

Boi, P.

T. Nakajima, G. Tonna, R. Rao, P. Boi, Y. Kaufman, B. Holben, “Use of sky brightness measurements from ground for remote sensing of particulate polydispersions,” Appl. Opt. 35, 2672–2686 (1996).
[CrossRef] [PubMed]

G. Dalu, R. Rao, A. Pompei, P. Boi, G. Tonna, B. Olivieri, “Aerosol optical properties retrieved from solar aureole measurements over southern Sardinia,” J. Geophys. Res. 100, 26,135–26,140 (1995).
[CrossRef]

Caroli, E.

C. Tomasi, E. Caroli, V. Vitale, “Study of the relationship between Ångström’s wavelength exponent and Junge particle size distribution exponent,” J. Clim. Appl. Meteorol. 22, 1707–1716 (1983).
[CrossRef]

Dalu, G.

G. Dalu, R. Rao, A. Pompei, P. Boi, G. Tonna, B. Olivieri, “Aerosol optical properties retrieved from solar aureole measurements over southern Sardinia,” J. Geophys. Res. 100, 26,135–26,140 (1995).
[CrossRef]

Deroo, C.

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

Deschamps, P. Y.

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

Di Bello, D.

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

Duhaut, P.

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

Esposito, F.

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

Fraser, R. S.

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

Ganor, E.

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

Gitelson, A.

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

Hayasaka, T.

Herman, B. M.

J. A. Reagan, K. J. Thome, B. M. Herman, “A simple instrument and technique for measuring columnar water vapor via near-IR differential solar transmission measurements,” IEEE Trans. Geosci. Remote Sens. 30, 825–831 (1992).
[CrossRef]

M. D. King, B. M. Herman, “Determination of the ground albedo and the index of absorption of atmospheric particulates by remote sensing. I. Theory,” J. Atmos. Sci. 36, 163–173 (1979).
[CrossRef]

Herman, M.

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

Holben, B.

Holben, B. N.

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

Karnieli, A.

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

Kasten, F.

Kaufman, Y.

Kaufman, Y. J.

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

King, M. D.

M. D. King, B. M. Herman, “Determination of the ground albedo and the index of absorption of atmospheric particulates by remote sensing. I. Theory,” J. Atmos. Sci. 36, 163–173 (1979).
[CrossRef]

Krishna Moorthy, K.

K. Krishna Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

Krishna Murthy, B. V.

K. Krishna Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

Mattoo, S.

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

Miyake, Y.

Morcrette, J. J.

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

Nair, P. R.

K. Krishna Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

Nakajima, T.

Nakanishi, Y.

Olivieri, B.

G. Dalu, R. Rao, A. Pompei, P. Boi, G. Tonna, B. Olivieri, “Aerosol optical properties retrieved from solar aureole measurements over southern Sardinia,” J. Geophys. Res. 100, 26,135–26,140 (1995).
[CrossRef]

Pavese, G.

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

Perbos, J.

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

Pompei, A.

G. Dalu, R. Rao, A. Pompei, P. Boi, G. Tonna, B. Olivieri, “Aerosol optical properties retrieved from solar aureole measurements over southern Sardinia,” J. Geophys. Res. 100, 26,135–26,140 (1995).
[CrossRef]

Rao, R.

Reagan, J. A.

J. A. Reagan, K. J. Thome, B. M. Herman, “A simple instrument and technique for measuring columnar water vapor via near-IR differential solar transmission measurements,” IEEE Trans. Geosci. Remote Sens. 30, 825–831 (1992).
[CrossRef]

Romano, F.

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

Sasamoto, K.

Serio, C.

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

Shiobara, M.

Tanaka, M.

Tanré, D.

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

Thome, K. J.

J. A. Reagan, K. J. Thome, B. M. Herman, “A simple instrument and technique for measuring columnar water vapor via near-IR differential solar transmission measurements,” IEEE Trans. Geosci. Remote Sens. 30, 825–831 (1992).
[CrossRef]

Tomasi, C.

C. Tomasi, E. Caroli, V. Vitale, “Study of the relationship between Ångström’s wavelength exponent and Junge particle size distribution exponent,” J. Clim. Appl. Meteorol. 22, 1707–1716 (1983).
[CrossRef]

Tonna, G.

Vitale, V.

C. Tomasi, E. Caroli, V. Vitale, “Study of the relationship between Ångström’s wavelength exponent and Junge particle size distribution exponent,” J. Clim. Appl. Meteorol. 22, 1707–1716 (1983).
[CrossRef]

Young, A. T.

Appl. Opt. (5)

IEEE Trans. Geosci. Remote Sens. (1)

J. A. Reagan, K. J. Thome, B. M. Herman, “A simple instrument and technique for measuring columnar water vapor via near-IR differential solar transmission measurements,” IEEE Trans. Geosci. Remote Sens. 30, 825–831 (1992).
[CrossRef]

Int. J. Remote Sens. (1)

D. Tanré, C. Deroo, P. Duhaut, M. Herman, J. J. Morcrette, J. Perbos, P. Y. Deschamps, “Description of a computer code to simulate the satellite signal in the solar spectrum: the 5S code,” Int. J. Remote Sens. 2, 659–668 (1990).
[CrossRef]

J. Appl. Meteorol. (1)

K. Krishna Moorthy, P. R. Nair, B. V. Krishna Murthy, “Size distribution of coastal aerosols: effects of local sources and sinks,” J. Appl. Meteorol. 30, 844–852 (1991).
[CrossRef]

J. Atmos. Sci. (1)

M. D. King, B. M. Herman, “Determination of the ground albedo and the index of absorption of atmospheric particulates by remote sensing. I. Theory,” J. Atmos. Sci. 36, 163–173 (1979).
[CrossRef]

J. Clim. Appl. Meteorol. (1)

C. Tomasi, E. Caroli, V. Vitale, “Study of the relationship between Ångström’s wavelength exponent and Junge particle size distribution exponent,” J. Clim. Appl. Meteorol. 22, 1707–1716 (1983).
[CrossRef]

J. Geophys. Res. (3)

Y. J. Kaufman, A. Gitelson, A. Karnieli, E. Ganor, R. S. Fraser, T. Nakajima, S. Mattoo, B. N. Holben, “Size distribution and scattering phase function of aerosol particles retrieved from sky brightness measurements,” J. Geophys. Res. 99, 10,341–10,356 (1994).
[CrossRef]

G. Dalu, R. Rao, A. Pompei, P. Boi, G. Tonna, B. Olivieri, “Aerosol optical properties retrieved from solar aureole measurements over southern Sardinia,” J. Geophys. Res. 100, 26,135–26,140 (1995).
[CrossRef]

U. Amato, D. Di Bello, F. Esposito, C. Serio, G. Pavese, F. Romano, “Intercomparing the Twomey method with a multimodal lognormal approach to retrieve the aerosol size distribution,” J. Geophys. Res. 101, 19,267–19,275 (1996).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Flow chart of the REDML program for inverting data of direct and diffuse solar irradiance.

Fig. 2
Fig. 2

Block diagram showing the procedure for data elaboration. In step 1 ΔΩ is a function of wavelength; in step 2 the R i , Θ j ) data represent all the available sets of data; ind., indicative value of.

Fig. 3
Fig. 3

Geometry of the solar disk scanning method.

Fig. 4
Fig. 4

Example of the dependence of R on Θ at λ = 0.369, 0.500, and 1.048 µm in the almucantar geometry. Data are from measurements with the Pom-01 radiometer.

Fig. 5
Fig. 5

Top, volume spectra retrieved with various values of m, k, and A. Bottom, relative differences (in percent) for the reference values listed. Data are from measurements with the radiometer Pom-01L. Mode INDM = 2, 3° ≤ Θ ≤ 30°.

Fig. 6
Fig. 6

Top, behavior of τ A with λ for various values of m, k, and A. Bottom, relative differences (in percent) for the reference values listed. Data are from measurements with the Pom-01L radiometer. Mode INDM = 2, 3° ≤ Θ ≤ 30°.

Fig. 7
Fig. 7

Top, dependence of ∊̅(R) at different hours on m, on k, and on A. Bottom, ∊̅(R) further averaged over time. Data are from measurements made with the Pom-01L radiometer. Mode INDM = 2, 3° ≤ Θ ≤ 30°.

Fig. 8
Fig. 8

Volume spectra retrieved with various radius intervals from experimental data measured with the Pom-01 radiometer. The number at the right in each figure is the value of the ∊̅(R) parameter.

Fig. 9
Fig. 9

Solar disk scanning: isolines of irradiance as functions of the row and column indices of the data matrix. Data are from measurements.

Fig. 10
Fig. 10

Same as Fig. 9 but for Ushuaia.

Fig. 11
Fig. 11

Behavior of the response function f(s) at the four wavelengths of the Pom-01L radiometer.

Fig. 12
Fig. 12

Optical depths for this plot were derived from R data (INDM = 2) with input parameter m* = 1.45 and A* values variable with time as in Table 5. Experimental data were taken from 15h45m to 18h00m with the Pom-01 radiometer.

Fig. 13
Fig. 13

Scatter diagram of points (τAdir and τAdiff); all wavelengths together. Data from Carloforte were taken from 13h00m to 17h45m and from Cagliari from 15h45m to 18h00m; radiometer Pom-01. The ground albedo is a function of wavelength and time.

Fig. 14
Fig. 14

Top, volume spectra v* and v**. Bottom, relative difference (in percent) between the above two spectra. Data were taken with the Pom-01 radiometer.

Fig. 15
Fig. 15

Top, behavior of τAdir and τAdiff with wavelength. Bottom, relative difference (in percent) between τAdir and τAdiff. Data were taken with the Pom-01 radiometer.

Fig. 16
Fig. 16

Left, given volume spectrum, retrieved volume spectrum expressed as a histogram (old) and as a set of log normals (new). Right, relative differences of the retrieved spectra against the given spectrum. Data are from a simulation.

Tables (7)

Tables Icon

Table 1 Mean Values of ΔΩ and Relevant rms Found from 15 Measurements of Solar Disk Scanning at Various Sites as a Function of Wavelength for the Pom-01L Radiometer

Tables Icon

Table 2 Behavior of ∊̅c with Aa

Tables Icon

Table 3 Behavior of V0 versus k and A at λ = 0.500 µm for the Same Data as in Table 2

Tables Icon

Table 4 Behavior of ∊(R1), … , ∊(R6) as a Function of Single Values for A for the Third and Last Iterationa

Tables Icon

Table 5 Values of A* for Measurements Taken at Cagliari on 19.7.95 over Several Hours from 15h45m to 18h00m with the Pom-01 Radiometera

Tables Icon

Table 6 Correlation Coefficient, Relative rms, and c Parameter as Functions of Wavelength for the Direct-Diffuse Langley Plot for the Data Taken at Cagliari on 19.7.95 Specified in Fig. 12a

Tables Icon

Table 7 Relative Differences (in %) between Values of V0* Found by the Direct-Diffuse Langley Plots and True Values (V0* ≡ 1) for the Four Days of Simulations

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

F=F0 exp-m0τ,
RΘEΘFm0ΔΩ=ωτPΘ+rΘβΘ+rΘ,
V=V0 exp-m0τ,
RΘVEVm0ΔΩ=βΘ+rΘ.
ΔΩ=ΔAEx, yE0, 0dxdy,
ΔΩ=2πβ  fssds,
ln V=ln V0-cm0τdiff

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